A note on some coupled fixed-point theorems on G-metric spaces

The purpose of this paper is to extend some recent coupled fixed-point theorems in the context of G-metric space by essentially different and more natural way. We state some examples to illustrate our results.

MSC: 46N40, 47H10, 54H25, 46T99.

In nonlinear functional analysis, one of the most productive tools is the fixed-point theory, which has numerous applications in many quantitative disciplines such as biology, chemistry, computer science, and additionally in many branches of engineering. In this theory, the Banach contraction principle can be considered as a cornerstone pioneering result which in elementary terms states that each contraction has a unique fixed point in a complete metric space. Due to its potential of applications in the fields above mentioned and many more, the fixed-point theory, in particular, the Banach contraction principle, attracts considerable attention from many authors (see, e.g., [4-30]). Especially, it is considered very natural and curious to investigate the existence and uniqueness of a fixed point for several contraction type mappings in various abstract spaces. A major example in this direction is the work of Mustafa and Sims [19] in which they introduced the concept of G-metric spaces as a generalization of (usual) metric spaces in 2004. After this remarkable paper, a number of papers have appeared on this topic in the literature (see, e.g., [1-8,10,12,18-29]).

For the sake of completeness, we recall some basic definitions and elementary results from the literature. Throughout this paper, is the set of nonnegative integers, and is the set of positive integers.

Definition 1 (See [19])

Let X be a nonempty set, be a function satisfying the following properties:

(G1) if ,

(G2) for all with ,

(G3) for all with ,

(G4) (symmetry in all three variables),

(G5) for all (rectangle inequality).

Then the function G is called a generalized metric, or more specially, a G-metric on X, and the pair is called a G-metric space.

Every G-metric on X defines a metric on X by

Example 2 Let be a metric space. The function , defined by

or

for all , is a G-metric on X.

Definition 3 (See [19])

Let be a G-metric space, and let be a sequence of points of X, therefore, we say that is G-convergent to if , that is, for any , there exists such that , for all . We call x the limit of the sequence and write or .

Proposition 4 (See [19])

Letbe aG-metric space. The following are equivalent:

(1) isG-convergent tox,

(2) as,

(3) as,

(4) as.

Definition 5 (See [19])

Let be a G-metric space. A sequence is called a G-Cauchy sequence if, for any , there is such that for all , that is, as .

Proposition 6 (See [19])

Letbe aG-metric space. Then the following are equivalent:

(1) the sequenceisG-Cauchy,

(2) for any, there existssuch that, for all.

Definition 7 (See [19])

A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .

Definition 8 Let be a G-metric space. A mapping is said to be continuous if for any three G-convergent sequences , and converging to x, y, and z, respectively, is G-convergent to .

Definition 9 Let and be mappings. The mappings F and g are said to commute if

In [27], Shatanawi proved the following theorems.

Theorem 10Letbe aG-metric space. Letandbe two mappings such that

Assume that F and g satisfy the following conditions:

(1) ,

(2) isG-complete,

(3) gisG-continuous and commutes withF.

If, then there is a uniquesuch that.

Corollary 11Letbe a completeG-metric space. Letbe a mapping such that

If, then there is a uniquesuch that.

In this paper, we aim to extend the above coupled fixed-point results.

We start with an example to show the weakness of Theorem 10.

Example 12 Let . Define by

for all . Then is a G-metric space. Define a map by and by for all . Then, for all with , we have

and

Then it is easy to that there is no such that

for all . Thus, Theorem 10 cannot be applied to this example. However, it is easy to see that 0 is the unique point such that .

We now state our first result which successively guarantee a coupled fixed point.

Theorem 13Letbe aG-metric space. Letandbe two mappings such that

for all. Assume thatFandgsatisfy the following conditions:

(1) ,

(2) isG-complete,

(3) gisG-continuous and commutes withF.

If, then there is a uniquesuch that.

Proof Take . Noting that , we can construct two sequences and in X such that

Let

Then, by using (2.1), for each , we have

which yields that

Now, for all with , by using rectangle inequality of G-metric and (2.2), we get

which yields that

Then, by Proposition 6, we conclude that the sequences and are G-Cauchy.

Noting that is G-complete, there exist such that and are G-convergent to x and y, respectively, i.e.,

Also, since g is G-continuous, we get

In addition, by (2.1) and the fact g commutes with F, we get

Combining this with (2.3), we get

On the other hand, by the fact that G is continuous on its variables (cf. [19]), we have

Thus, we conclude that

i.e.,

which yields that

Moreover, it follows from

that . Thus, , i.e., .

Next, let us show that . By using rectangle inequality of G-metric and (2.1), we have

which gives that

Combing this with the fact that and are G-convergent to x and y, respectively, we conclude that

which yields that

Recalling that and , we get and .

It remains to show the uniqueness. Let be such that . Then we have

which yields that . Thus, , which means . This completes the proof. □

Remark 14 It is easy to see that Theorem 10, appearing in [27], is a direct corollary of Theorem 13. On the other hand, Theorem 13 can deal with some cases, which Theorem 10 cannot be applied. For this, let us reconsider Example 12. In fact, for all , we have

i.e., (2.1) holds. Other assumptions of Theorem 13 are easy to verify. So, by Theorem 13, there exists a unique such that .

Letting , we can get the following result.

Corollary 15Letbe a completeG-metric space. Letbe a mapping such that

for all. If, then there is a uniquesuch that.

Example 16 Let be the same as in Example 12. Then is a G-metric space. Also, it is not difficult to verify that is G-complete. Define a map by for all . Then, for all , we have

and

Thus, the statement (2.4) of Corollary 15 is satisfied for any . Thus, there is a unique such that .

Remark 17 Corollary 11 cannot be applied to Example 16 since (1.3) does not hold. In fact, if (1.3) holds for some , then

which is a contradiction.

The authors declare that they have no competing interests.

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

The authors are indebted to the referees for their careful reading of the manuscript and valuable suggestions. Hui-Sheng Ding acknowledges support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), the Jiangxi Provincial Education Department (GJJ12173), and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University.

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